ASTER THESIS - Bibsys Yue... · 2016-12-16 · In this thesis project, a numerical model based on Volume of Fluid ... Volume of Fluid (VOF), free surface tank (FST), model tests,

MASTER THESIS

Postal adress: Visit adress Telephone Fax Bank Høgskolen i Ålesund Larsgårdsvegen 2 70 16 12 00 70 16 13 00 7694 05 00636 N-6025 Ålesund Internett E-mail Enterprise no. Norway www.hials.no [email protected] NO 971 572 140

TITLE:

Computional Fluid Dynamics(CFD) Study on Free Surface Anti-Roll Tank and

Experimental Validation

CANDIDATE NAME:

Yue Li

DATE: COURSE CODE: COURSE TITLE: RESTRICTION:

29/05/2015 IP501909 Master thesis

STUDY PROGRAM: PAGES/APPENDIX: LIBRARY NO.:

Master in ship design 65/10

SUPERVISOR(S):

Karl Henning Halse

ABSTRACT:

The excessive motion of a ship can seriously degrade the performance of machinery and

personnel. Anti-roll tanks are tanks fitted onto ships in order to improve their response to roll

motion, which is typically the largest amplitude of all the degrees of freedom. In this thesis

project, a numerical model based on Volume of Fluid (VOF) is introduced to simulate the liquid

sloshing inside a free surface tank (FST). And this particular model is suitable for a general

thermostatic incompressible sloshing problem.

Through the simulation of the model, the performance of the FST is fully investigated when the

tank is excited by different periods and different amplitudes. Besides, different tank filling

levels are studied to find out the optimal filling level. The results are proved by the existing

experimental database.

Furthermore, a model platform is designed and established to give further validation to the

numerical model. The comparisons between the model tests and simulations have shown good

consistency. Further improvements are proposed in the end.

Keywords: Volume of Fluid (VOF), free surface tank (FST), model tests, validation

This thesis is submitted for evaluation at Ålesund University College.

mailto:[email protected]

AALESUND UNIVERSITY COLLEGE PAGE 2

PREFACE

This master thesis was written at the Aalesund University College, Faculty of Maritime

Technology and Operations (AMO), to earn the degree Master of Science.

The present work constitutes the graduation in the 2 Years International Master Programme in

Ship design. The thesis has been made solely by the author; some contents used in the

project, however, is based on previous works of others. Through reference has been made

accordingly.

The thesis project contains CFD simulations as well as physical model tests and verification.

Experimental platform design, establishment and tests are also part of the thesis.

My special thanks go to Associated Prof. Karl Henning Halse who was willingly supervising me.

I appreciate his guidance and support throughout the entire thesis composition. The new ideas

inspired and new fields of knowledge introduced by him has always been fruitful to me.

Thanks to Geir Åge Øye from Rolls-Royce Marine AS for sharing his experience within this field.

I would also like to thank Professor Vilmar Ærøy, Phd candidate Grotle Erlend Liavåg for their

assistance on platform design and establishment. Thank Stig Morten Skaar, Nikolas

Myklebostad, Anders Sætersmoen and Wei Li for assisting the model platform manufacturing

and running.

No less thanks to my parents who enabled my studies and supported me in many respects.

Thanks to my friends who made the time here in Norway so special.

Aalesund University College

May. 2015

Yue Li


ABSTRACT


personnel. Anti-roll tanks are tanks fitted onto ships in order to improve their response to roll

motion, which is typically the largest amplitude of all the degrees of freedom. In this thesis

project, a numerical model based on Volume of Fluid (VOF) is introduced to simulate the liquid

sloshing inside a free surface tank (FST). And this particular model is suitable for a general

thermostatic incompressible sloshing problem.

Through the simulation of the model, the performance of the FST is fully investigated when the

tank is excited by different periods and different amplitudes. Besides, different tank filling

levels are studied to find out the optimal filling level. The results are proved by the existing

experimental database.

Furthermore, a model platform is designed and established to give further validation to the

numerical model. The comparisons between the model tests and simulations have shown good

consistency. Further improvements are proposed in the end.

Keywords: Volume of Fluid (VOF), free surface tank (FST), model tests, validation


Table of contents

TERMINOLOGY ............................................................................................................................................................. 5

SYMBOLS ......................................................................................................................................................................... 5 ABBREVIATIONS .............................................................................................................................................................. 5

1 INTRODUCTION .................................................................................................................................................... 6

1.1 MOTIVATION AND OBJECTIVE ............................................................................................................................. 6 1.2 THESIS STRUCTURE .............................................................................................................................................. 7

2 BACKGROUND AND THEORETICAL BASIS .................................................................................................. 8

2.1 MECHANISM OF ANTI-ROLL TANKS ...................................................................................................................... 8 2.2 DEVELOPMENT OF SHIP ANTI-ROLL TANKS ........................................................................................................ 10 2.3 NUMERICAL METHODS ....................................................................................................................................... 13 2.4 KEY PARAMETERS OF FREE SURFACE TANK ....................................................................................................... 13

3 METHODS .............................................................................................................................................................. 18

3.1 GOVERNING EQUATIONS .................................................................................................................................... 18 3.2 VOF MODEL ..................................................................................................................................................... 18 3.3 TURBULENCE MODELLING ................................................................................................................................. 20

4 CFD SIMULATION ............................................................................................................................................... 22

4.1 COMPUTATIONAL MODEL .................................................................................................................................. 22 4.2 GRID (MESH) ..................................................................................................................................................... 24 4.3 TEMPORAL DISCRETIZATION SCHEME ................................................................................................................ 25 4.4 TIME STEP CONTROL .......................................................................................................................................... 26 4.5 TURBULENCE MODELLING ................................................................................................................................. 28 4.6 2D VS 3D MODEL ............................................................................................................................................... 32

5 TANK PERFORMANCE DISCUSSION ............................................................................................................. 35

5.1 MOMENT AND PHASE ......................................................................................................................................... 35 5.2 NATURAL PERIOD .............................................................................................................................................. 35 5.3 EXCITATION AMPLITUDE ................................................................................................................................... 42 5.4 FILLING LEVEL/PERCENTAGE ............................................................................................................................. 44 5.5 COVER PANEL EFFECT ....................................................................................................................................... 45

6 EXPERIMENT AND VALIDATION ................................................................................................................... 48

6.1 EXPERIMENTAL DATA FROM SHIPX ................................................................................................................... 48 6.1.1 Tank design in ShipX ................................................................................................................................ 48 6.1.2 Tank database ........................................................................................................................................... 48 6.1.3 Data comparison ....................................................................................................................................... 49

6.2 MODEL TEST SET UP AND PLAN ......................................................................................................................... 52 6.2.1 Model Tank Design ................................................................................................................................... 52 6.2.2 Platform Set Up ........................................................................................................................................ 52 6.2.3 Test Preparation and Plans ...................................................................................................................... 55 6.2.4 Data post-processing ................................................................................................................................ 56

7 DISCUSSION .......................................................................................................................................................... 57

7.1 COMPARISON BETWEEN MODEL TESTS AND SIMULATIONS ................................................................................. 57 7.2 CHALLENGES IN THIS PROJECT ........................................................................................................................... 61

8 CONCLUSIONS ..................................................................................................................................................... 62

REFERENCES ............................................................................................................................................................... 63

APPENDIX...................................................................................................................................................................... 65


TERMINOLOGY

Symbols

A Excitation amplitude [deg]

b Tank width [m]

𝐶𝑁 Courant number [-]

h water level [m]

h/b Tank filling ratio [-]

H Tank height [m]

i Mode number [-]

Lpp Length between perpendiculars [m]

M Damping moment [Nm]

t Time step [s]

𝑇𝑛 Natural period [s]

𝑢𝑖 Flow velocity [m/s2]

𝑥𝑖 Grid spacing at node i [m]

Abbreviations

ART Anti-roll tank

CFD Computational Fluid Dynamics

FST Free surface tank

LNG Liquid Natural Gas

RAO Response Amplitude Operator

RSM Reynolds-stress model

SST Shear Stress Transport

VERES ShipX Vessel Responses program

VCG Vertical center of gravity

VOF Volume of Fluid


1 INTRODUCTION

In former times ships were stabilised in the wind by their huge sails and large keels.

When ships began sailing with motor power they started to use bilge keels but these

were not sufficient enough to compensate for the upcoming roll motions. It was already

in 1880 when Watts and Froude suggested the use of water tanks for damping as

reported by (Bosch 1966). Nowadays, roll stabilizing tanks are widely used on offshore

supply vessels and fishing vessels to provide a better and safer working environment.

Tanks of different shape exist and they can be passively or actively controlled.

The movement of water within a container is called sloshing. Sloshing appears in almost

every moving vessel which contains liquids with a free surface in a partially filled tank.

(O. T. Faltinsen 2009) pointed out that the liquid will move severely from one side to the

other as a result of resonant excitation motions from the ship. As examined later,

sloshing has a strong influence on the dynamic stability of a vessel.

Ship motions excited by waves can evoke sloshing in a tank which in turn will influence

the ship motions from inside the ship. Ships equipped with a specially tuned tank utilize

this effect to dampen their roll motions.

1.1 Motivation and Objective Since the 1960 s first liquid natural gas (LNG) tankers were built to serve Europe and

Japan with a new carrier of energy. It became apparent that sloshing in cargo tanks

resulted in one of the most critical ship loads. Until today these generated loads have

remarkable influence on the tank walls and the supporting ship structure as reported by

(O. M. Faltinsen 1974) and (Solaas 1995). Spherical and membrane tanks are the two

mainly used geometries. A failure of such a tank can lead to brittle fracture of the

surrounding structure due to a low temperature shock of the escaping liquid. Besides the

risk of explosion the repair costs and out-of-service costs are high.

The phenomenon of sloshing appears not only in ships but also in many different

applications and vehicles. Recently sloshing in seismically excited tanks is investigated

that (Goudarzi 2012). Other storage tanks are installed on floating offshore units for oil

and gas production. Tanks of well boats contain living fish which has to be transported

carefully to fish farms. Solaas (1995) mentioned sloshing in railway tanks. One can also

find many articles written about sloshing in fuel tanks of rockets.

As can be seen from the described cases, cargo tanks apply manifold and sloshing

becomes an important aspect. According to Solaas (1995), sloshing can involve large

fluid motions with turbulences, braking waves and spray. These are highly nonlinear

phenomenon and sloshing itself is therefore difficult to predict.

Besides the undesired sloshing in cargo tanks, sloshing is required in roll damping tanks

to a certain amount. Still, many model tests are conducted to find the best configuration

for a roll damping tank. On the other hand, recently studies have tried to perform

numerical simulations of the complicated 3D motion of water in tanks together with the

sloshing at the free surface. Different numerical methods have been tested by

researchers.

This report considers the suitability of a commercial Navier Stokes Computational Fluid

Dynamics software for the simulation of free surface tank. As CFD is a relatively new

method of simulating fluid dynamics problems, there is as yet no consensus as to the

most suitable approach. Given the assortment of possible solution strategies available, it

is necessary to study the influence of user choice on the accuracy of the obtained

solution. This report seeks to establish the significance of model-specific computational

parameters and develop guidelines for future CFD sloshing models. Besides,

computational solutions are compared to experimental tests at Aalesund University

College. In this study the following are investigated:

Summary of current ship anti-roll tanks;


Grid (mesh) independence;

Time marching scheme;

Turbulence modelling (K- Epsilon & K-Omega);

2D flow and 3D flow;

Natural period of the tank;

Tank performance at different excitation amplitudes;

Tank performance at different filling levels;

Optimal filling level;

The influence of cover panel.

1.2 Thesis structure In the following chapters the theoretical aspects of a free surface tank will be discussed

first. This will be done by describing the mechanism of the tank and how this kind of tank

works. Then it summarized the current anti-roll tank designs and the numerical methods

to this particular problem.

The second part of the report will focus on the CFD simulations. It aims to develop

guidelines for a generic anti-roll tank simulation model. Therefore, major considerations

are discussed including the mesh independence, time marching scheme, time step

control, turbulence modelling and 2D/3D flow difference. And this particular model is

suitable for a general thermostatic incompressible sloshing problem.

Chapter 4 will study the the anti-roll tank performance. Firstly, the natural period of the

partially filled tank is investigated and discussed. Then different excitation amplitudes will

be studied. Tank performance at different filling levels are also of interest. Besides, a

small investigation on cover panel effect is included.

The last part of the report will focus on the model tests. It will compare the results from

simulations with the exsiting experimental database. The model tests, which designed

and performed by the author, have been used to provide validations for the numerical

simulations as well.


2 BACKGROUND AND THEORETICAL BASIS

The subject of roll stabilisation has been investigated for many years by Naval Architects

and designers. The main reason for this continued interest is that for roll, unlike pitch or

other degrees of freedom, the forces required to create a stabilising effect are relatively

small. So far, different stablizing devices have been designed such as Bilge keel, fin

stabilizer and anti-roll tank. Figure 1 shows the typical Roll response of a ship with and

without a passive roll damping tank.

Figure 1. Roll response of a ship with and without a passive roll damping tank (Lewis 1989)

2.1 Mechanism of anti-roll tanks

Figure 2. Wave patterns inside a free surface tank

The mechanism by which the flowing water can affect a vessel motion is a combination of

various wave phenomena, as shown in Figure 2. Firstly, and most importantly, the

variation in water mass from one side of the tank to the other can create an additional

moment about the roll-centre, the phase difference between the motion of the water and

the ship roll determining whether the effect is to increase or reduce the roll amplitude.

Secondly the turbulent flow of water within the tank can be used to reduce roll by


removing kinetic energy from the ship’s motion. This is achieved by the formations of

hydraulic jumps and turbulence. The creation of hydraulic jumps, or wave bores, within

the tank at certain depths and roll frequencies can also result in a considerable impulse

when the bore hits the tank wall. This, coupled with the mass of water moving

harmonically across the tank, can have a significant effect on the ship motion.

There are a great number of roll stabilisation devices available for use on vessels today.

Many of these were initially developed for use on large ships but are increasingly being

installed on smaller vessels where the subsequent benefit to the vessel’s motions is often

more apparent. This is due to the relatively high transverse stability, and short natural

roll periods of smaller vessels (Martin 1994). The prevalence of shorter waves also

means that smaller vessels are more likely to encounter resonant wave conditions and

thus violent rolling.

Hydraulic jump

Due to roll motions of the ship, the fluid in the tank will flow from one side to the other

across the tank. The shifting mass will exert an own roll moment on the ship. By a smart

design of the tank this roll moment can be used as a damping moment which counteracts

the initial roll motion of the ship as stated by (Moaleji 2007). Thereby a so called

hydraulic jump occurs when the lowest natural frequency of the tank is near the

excitation frequency. The flow motion with a hydraulic jump is schematically illustrated

by Faltinsen and Timokha (2009) in Figure 3.

Figure 3. Psition of the hydraulic jump during one period of roll at tank resonance

Standing wave

Henderson (2014) pointed out that waves can be reflected in a way such that the water

seems to stand still at certain locations in the tank while the surrounding medium is

moving up and down. These nodal points are illustrated in Figure 4 and a so called

standing wave pattern is formed.

Nodal and antinodal vertical lines pass through the liquid volume. A liquid particle moves

only horizontally at a nodal line, whereas the motions are only vertical at an antinodal

line. The lowest natural mode (i=1) has a node in the middle of the tank and antinodal


lines coinciding with the vertical walls. The number of nodal lines is equal to the mode

number, i. The nodes divide the interval [−1

2b, +

1

2b] into i+1 subintervals.

Figure 4 illustrates that the motion of a liquid particle associated with small-amplitude

freestanding waves by a natural mode is rectilinear, which differs from linear harmonic

long-crested propagating wave.

Figure 4. Standing waves corresponding to natural modes for sloshing in rectangular tank

2.2 Development of ship anti-roll tanks

2.2.1 Free surface tank (FST)

In 1875 Watts first introduced the free surface anti-roll tank as a passive method of

damping rolling motions. He proposed a large, uniform cross section tank partly filled

with water, placed athwartships and usually located well above the centre of gravity, see

Figure 5. The principle was based on the work by (Froude 1861) who was the first to

frame the effect of waves on the rolling motion of ships.

Figure 5. Ideal motion of water in the tank

More recent experimental work was done by (Ikeda 1991) in which the coupling effect on

the performance of a rectangular anti-rolling tank was investigated. The results of the

bench test of an anti-rolling tank including the effect of sway motion as well as roll

motion demonstrated that the sway motion reduced the reduction of the roll angle by the

anti-rolling tank and lengthened the natural period of the tank.

It was observed that the performance of free-surface tanks is maximized when the

natural frequency of the tank is tuned to be close to the roll natural frequency of the

ship. This is mainly done by altering the water level inside the tank. This confirmed the

findings of Watts and Froude from a century earlier. Different responses of tanks can be


obtained by changing their shapes, two modifications of such tanks are presented in

Figure 6.

Figure 6. Plan view of modified tanks (a) C-shape tank and (b) rectangular tank with baffles

One disadvantage of rectangular tanks is that it is difficult to control the water, rushing

freely from side to side in the tank, threatening the safety of the ship in rough weather.

Sometimes a limited control is exerted over the motion of the fluid by installing a

restriction or baffle in the center of the tank. Lee and Vassalos (1996) experimentally

showed that the performance characteristics of an anti-roll tank could be ‘tailor-made’ by

a judicious application of flow obstructions.

2.2.2 U-shape tank

The simple free surface tank has three practical problems; two due to the sloshing of the

large free surface and the third to its location. Sloshing makes it difficult to control the

water and adversely affects stability while the tanks required location, above the centre if

gravity and near the centre of the ship, is prime usable space. These problems were

solved simultaneously with the U-shaped tank suggested by Frahm (1911).

Figure 7. Passive U-shape tank

It was found that for ships with a small metacentric height operating in seas with a

regular wave pattern, good results were obtained with a roll reduction of approximately

50%. However, in choppy seas with no regular pattern, the results were poor, frequently

with no observed roll reduction at all (Bennett 1991).

Figure 8. Activated U-shape tank

A logical development of the Frahm system was to pump the water from one leg of the U

to the other, rather than relying on the ship’s rolling motion exclusively for the transfer.

By doing so, a limitation of the passive tanks with their dependence on resonance could

be largely overcome. Such tanks are generally known as U-shape activated tanks or


simply ‘‘active tanks’’. A system was evolved by Bell and Walker (1966), which achieved

more than the damping effect of a passive-type tank and also did not involve setting into

motion a long column of fluid. Fig. 8 shows the arrangement of this system. It is seen

that the impeller which rotates continuously drives fluid from the lower to the upper of

the two ducts connecting the port and starboard tanks.

2.2.3 Free-flooding tank

In some installations, the horizontal leg of the U-tank was entirely removed and the

bottoms of the tanks were opened to the sea, Figure 9 (left). One of the few cases of

installation of free-flooding tanks was in 1931 and 1932 when tanks were retrofitted to

six USN cruisers of the Pensacola and Northampton classes. The tanks had no air cross

connection, despite initial misgivings the tanks were successful reducing the roll motion

by 30–40% and increasing the roll period by 20%.

Figure 9. Free-flooding tank

In 1958 a more advanced version of free-flooding tanks were tested on a model in the

Denny Model Ship Tank at Stevens Institute of Technology (Figure 9 right).

Results showed that this system could reduce roll by about 60% (Vasta 1961). The

effectiveness of the free-flooding tanks falls off as the ship speed increases, because at

higher speeds, water cannot get into the flooding port. The main advantage of free-

flooding tanks is that they do not need a water crossover duct, and therefore do not

require major modifications of the ship when installation is contemplated in an existing

ship. Free-flooding tanks use seawater as working fluid and this can have considerable

impact on the design in terms of corrosion, fouling and maintenance.

2.2.4 Summary of anti-roll tanks Table 1. Summary of different anti-roll tanks

Type Control Roll reduction Merit Defect

Free surface tank

Passive 37.5% Simple; cost efficient

Free liquid surface Difficult to control the water and

adversely affects stability；

Location

Rectangular tank with baffles

Passive up to 80% in periodic waves; up to 50% in irregular waves

easily be adapted to another condition of loading changing the water depth

Location

Passive U-tank

Passive/ Passive air control

Approximately 50%

Good performance in seas with a regular wave pattern

Poor performance in choppy seas with no regular pattern; Water crossover duct

Activated U-tank

Activated (Same as passive U tank)

- Power consumption; Water crossover duct

Free-flooding tank

Activated 60% No water crossover duct Water cannot get into the flooding port at higher speeds.


2.3 Numerical methods Recently studies have tried to perform numerical simulations of the complicated 3D

motion of water in tanks together with the sloshing at the free surface, both for simple

open tanks and U-tanks. A specific numerical study using a 2D finite element method for

U-tube passive anti-rolling tanks was done by Zhong et al. (1998) who reported

numerical analysis of sloshing in a road container with a MAC type method for tracing the

free surface evolution. Many papers have reported on the sloshing phenomenon in a rigid

rectangular tank and theoretical and experimental research has been carried out by

means of the shallow water wave theory (Chern et al., 1999; boundary element methods

by Liu and Huang, 1994; finite element methods, VOF method by Celebi and Akyildiz,

2002. Extensive comparative studies on sloshing loads has been made by Sames et al.,

2002. Most of the methods reported were based upon finite difference, finite volume

approaches or even shallow water equation solvers.

Another interesting solution was proposed by Souto Iglesias et al. (2004). They

simulated the motion of water using the smoothed particle hydrodynamics method where

the fluid is represented by a set of particles which follow the fluid motion and have the

fluid quantities such as mass and momentum. Associated with each particle are its

position, velocity, and mass. They validated their method by performing experiments on

different kind of free-surface tanks such as rectangular tanks with and without baffles

and on a C-tank, and showed very good agreement between the simulation and

experimental results.

2.4 Key parameters of free surface tank

2.4.1 Tank dimensions

Figure 10. Definition of geometry and tank dimensions: Free surface tank (FST)

The following parameters are important when the tank moment and phase is to be

determined:

Tank geometry/dimensions: Defined by the width b, length L and height H.

Vertical position relative to the rotation axis, s/b: Characterized by the vertical

distance from the tank bottom to the ship’s center of gravity, s. The factor s/b is

often used to describe this parameter, where s is the vertical distance defined as

positive when the tank bottom is above COG.

Tank filling, h: Denoted by the non-dimensional number h/b where h is the fluid

level in the tank.

In practice, for shallow-liquid sloshing, tank filling rations ℎ/𝑏 ≤ 0.05 − 0.1. The

reason for the lower range is that it is difficult to obtain good effect of the tank for

lower fillings (large roll angles leads to a partly dry tank bottom). 0.05 − 0.1 ≤ ℎ/𝑏 ≤


0.2 − 0.25 is so-called intermediate depths. Finite depth is a depth larger than the

intermediate depth and corresponds to when 𝑡𝑎𝑛ℎ(𝜋ℎ/𝑙) is not close to 1. Finally, the

deep-water approximation implies relatively large h/l so that 𝑡𝑎𝑛ℎ(𝜋ℎ/𝑙) ≈ 1.

2.4.2 Natural period

Natural frequencies or eigenfrequencies describe the modes in which a body or fluid can

oscillate when excited. The modes are characteristic wave patterns as shown in Figure 4.

The relationship of any period T and frequency ω is

𝑇 =2π

ω

The natural sloshing frequencies and periods for a roll damping tank with arbitrary water

depth are given by

𝜎𝑛 = √π∙i

𝑏𝑔 ∙ 𝑡𝑎𝑛ℎ (

π∙i

𝑏∙ ℎ) (1)

𝑇𝑛 = 2π/√π∙i

𝑏𝑔 ∙ 𝑡𝑎𝑛ℎ (

π∙i

𝑏∙ ℎ) , 𝑖 = 1,2, … (2)

Where

i is mode number;

b is tank breadth;

g is gravity constant;

h is water depth;

T1 represents the highest natural period since increasing i in the denominator of the

fraction gives lower period values.

From the above equation, it can be clearly seen that the water depth h and the tank

breadth b have great influence on the natural tank period. So, the performance of the

given tank can be adjusted to different roll periods due to changing loading conditions,

by simply regulating the water depth in the tank. The following figure 11 based on

equation (2) illustrates that the natural period increases with increasing tank breadth b

for a given filling level h. For a given breadth b the natural period will decrease for

increasing h.

Figure 11. Natural sloshing period for a rectangular tank versus the tank breadth b for different filling levels (Faltinsen, 2009)


2.4.3 Hydrostatic and Dynamic pressure

The relationship of wave length and tank breadth is given as:

𝑛 𝜆 = 2 𝑏 (3)

According to this relationship, the typical wave patterns for the natural periods can be

drawn as in figure 12. For n = 1 for the first natural period in the upper part of the

figure, the longest possible wave is given whereby only half of the wave can be seen

within the tank. There is most of the water concentrated on one side of the tank which

gives high dynamic and hydrostatic pressure forces resulting in a high moment. The first

natural period gives therefore the highest damping moment when the tank is installed on

a ship. In the following, the highest natural period is also referred to as the resonance

period as excitation at this period leads to resonance.

Figure 12. Pressure in a FST

For the second natural period on the right in figure 12, the forces stay in equilibrium and

give no damping moment. There is one whole wave in the tank but with a lower

amplitude than for the first, highest natural period. The lower the periods or higher the

frequencies get, the more waves with an even lower amplitude will be in the tank as

exemplified in the lower part of the figure.

2.4.4 Damping

Damping is given when the moment is negative to the velocity as illustrated in Figure 3.

The tank is moving with a roll angle 𝜂4:

𝜂4 = 𝜂4𝑎 ∙ sin (𝜔𝑡) (4)

The measured moment is

𝑀 = 𝑀𝑎 ∙ sin (𝜔𝑡 − 𝜑) (5)

with a phase shift 𝜑 between roll angle and moment. The moment can be split into

𝑀 = 𝑀𝑎 ∙ 𝐴 ∙ sin(𝜔𝑡) + 𝑀𝑎 ∙ 𝐵 ∙ cos (𝜔𝑡) (6)

The first part after the equal sign of equation (6) represents a part of the excitation force

which is needed to give a forced motion on the tank. The sine components are inertia

and restoring forces as in the equation of motion (8) for roll which applies for a tank in

combination with a ship:

𝑚𝜂4 + 𝑐𝜂4 + 𝑘𝜂4 = 𝑃4 (7)

−𝜔2𝑚𝜂4𝑎 𝑠𝑖𝑛 (𝜔𝑡) + 𝜔𝑐 𝜂4𝑎 𝑐𝑜𝑠(𝜔𝑡) + 𝑘 𝜂4𝑎 𝑠𝑖𝑛 (𝜔𝑡) = 𝑃4𝑎 𝑠𝑖𝑛(𝜔𝑡 − 𝜑) (8)

In equation (8), inertia forces plus damping forces plus restoring forces equal to the

excitation forces which can be phase shifted to the other terms. The factor B in the last


part of equation (6) finally is the damping component of the moment in the tank as this

cosine term is in phase with the cosine term of the equation of motion (8) where this part

represents the damping in the system. There is a 90° phase difference between the tank

motions and the stabilizing moment because the cosine term of the moment is 90° phase

shifted to the sine term in equation (4) for the tank motions. By adding a stabilizing

moment to the unstable roll motion the roll motion will become well stabilized as on the

last line of figure 13.

Figure 13. Achieving 90° phase difference for a stabilized motion (Winkler 2012)

A system will oscillate with the same period as the excitation force which can be a

harmonic function as in equation (8). The amplitude of the displacement can be found by

𝑎 = 𝑃𝑎/𝑘 ∙ 𝐷𝐿𝐹 (9)

Here, Pa / k is the static displacement which will occur when the system is hidden with a

load amplitude Pa. The dynamic load factor (DLF) can enlarge or reduce the displacement

as the DLF is depending on the frequency ratio β and the damping ratio ξ. The frequency

ratio β is defined as the ratio between the load frequency ω and the eigenfrequency ω0:

𝛽 = 𝜔/𝜔0 (10)

The damping ratio ξ is defined as the ratio between the damping coefficient c and the

critical damping ccr:

𝜉 = 𝑐/𝑐𝑐𝑟 (11)

For a damped system in forced oscillation as it is the case for a roll damping tank, the

DLF becomes

𝐷𝐹𝐿 = 1/√(1 − 𝛽2)2 + (2𝜉𝛽)2 (12)

The dependency on both the frequency ratio β and the damping ratio ξ can be seen.

Formula 12 can be plotted as in figure 14 where the DLF varies for different frequency

ratios and for different damping ratios which are given in percent on the right hand side

of the figure where the order of the numbers complies with the order of the lines. This

figure points out that the response of a system will be very large when the load

frequency equals the eigenfrequencies such that the frequency ratio equals one. This

case is called resonance. For lower load frequencies than the eigenfrequencies, the DLF

approaches one which means that here the displacement of the system will follow the

slow excitation in phase. A so called static condition is reached. At the other end the

graph approaches zero because the excitation frequency is too high for the mass of the


system to follow. For this reason it is an inertia dominated system in that range. A high

load frequency gives then only small response amplitudes. As Larsen (2014) concluded,

the damping of a system is only for the resonance range of importance.

Figure 14. Dynamic load factor as function of the frequency ratio for given damping ratios

Corresponding to this issue, figure 15 exemplifies how large the phase shift between load

and response will be for different frequency ratios and for various damping. For a

frequency ratio smaller than one, load and response are almost in phase for low damping

and the term “quasi static response” is used. At resonance the response is 90° phase

shifted to the excitation. Thereafter both are opposite in phase for a frequency ratio

greater than one.

The phase shift is depending on the load frequency but not on the load s amplitude. The

phase shift is also depending on the damping in figure 15 as well as the response

amplitude in figure 14.

Figure 15: Phase angle between load and response as function of the frequency ratio for given

damping ratios (Palm 2007)


3 METHODS

3.1 Governing equations The Navier-Stokes Equations describe the behavior of a viscous (usually Newtonian)

fluid. This considerably more general model introduces nonlinearities rendering a closed-

form solution for all but the simplest cases impossible. Unfortunately sloshing is not one

of these.

In marine field, we normally neglect the energy part which assumes that the temperature

in our case is constant.

The motion of a fluid can be described by a set of partial differential equations expressing

conservation of mass, momentum and energy per unit volume of the fluid. The Navier

Stokes equations for three dimensional incompressible fluid flow can be written in

conservation form as follows

(13)

(14)

where

• 𝑣 is the velocity field vector;

• p is the pressure;

• ρ is the mass density;

• ν is the kinematic viscosity;

• g is the acceleration of gravity;

• 𝑒3 is the unit vector pointing upwards;

• ∇ is the gradient operator;

• ∇⋅ is the divergence operator;

• ∇⋅∇ is the Laplace-operator;

3.2 VOF Model Volume Of Fluid (VOF) is a simple multiphase model. It is suited to simulating flows of

several immiscible fluids on numerical grids capable of resolving the interface between

the phases of the mixture, as shown below. In such cases, there is no need for extra

modeling of inter-phase interaction, and the model assumption that all phases share

velocity, pressure, and temperature fields becomes a discretization error.

Figure 16: Illustration of unsuitable grids (left) and suitable grid (right) for two-phase flows using the VOF model


Due to its numerical efficiency, the model is suited for simulations of flows where each

phase constitutes a large structure, with a relatively small total contact area between

phases. A good example of this type of flow is sloshing flow in a water tank, where the

free surface always remains smooth. If the tank movement becomes pronounced, this

results in breaking waves, large numbers of air bubbles in the water, and water droplets

in the air. The method would then require a fine mesh (at least three cells across each

droplet/bubble) to produce small modeling errors.

Basic VOF Model Equations

The VOF model description assumes that all immiscible fluid phases present in a control

volume share velocity, pressure, and temperature fields. Therefore, the same set of basic

governing equations describing momentum, mass, and energy transport in a single-

phase flow is solved. The equations are solved for an equivalent fluid whose physical

properties are calculated as functions of the physical properties of its

constituent phases and their volume fractions.

The equations are:

(15) (2099)

(16) (2100)

(17) (2101)

where:

𝑎𝑖 = 𝑣𝑖/𝑣 is the volume fraction and 𝜌𝑖 , 𝜇𝑖 and ( 𝑐𝑝 )𝑖 are the density, molecular

viscosity and specific heat of the i th phase.

The conservation equation that describes the transport of volume fractions 𝑎𝑖 is:

(18)

(2102)

Where 𝑆𝑎𝑖 is the source or sink of the i th phase, and 𝐷𝜌𝑖/𝐷𝑡 is the material or

Lagrangian derivative of the phase densities 𝜌𝑖.

If there is a large time variation of phase volume fractions 𝑎𝑖 , there is a large time

variation of the mixture density 𝜌 which features in the continuity equation, Eqn. (18).

Since this unsteady term cannot be linearized in terms of pressure and velocity, it acts as

a large source term which can be “unpleasant” for a numerical treatment within a

segregated solution algorithm. Therefore, Eqn. (18) is rearranged in the following, non-

conservative form:

(19)

(2104)

In the case when phases have constant densities and have no sources, the continuity

equation reduces to .


3.3 Turbulence modelling There is no consensus apparent whether and when sloshing flow should be modelled as

turbulent or laminar. Previous studies hace shown that turbulence model is more widely

used. Rhee (2005) observed significant variations between laminar and turbulent flow

models in the test cases and concluded that turbulence effects should be taken into

account in a CFD model for this particular case. Similar effects have been observed in the

sensitivity study by Godderidge et al. (2006). Thus, we assume the flow in our cases to

be turbulent. A breif introduction on different turbulence models is shown below.

3.3.1 Standard K- Epsilon

A K-Epsilon turbulence model is a two-equation model in which transport equations are solved for the turbulent kinetic energy 𝑘 and its dissipation rate 𝜀 .

Turbulent dissipation is the rate at which velocity fluctuations dissipate. This is the

default k–ε model. Coefficients are empirically derived; valid for fully turbulent flows

only. In the standard k-e model, the eddy viscosity is determined from a single

turbulence length scale, so the calculated turbulent diffusion is that which occurs only at

the specified scale, whereas in reality all scales of motion will contribute to the turbulent

diffusion. The k-e model uses the gradient diffusion hypothesis to relate the Reynolds

stresses to the mean velocity gradients and the turbulent viscosity. Performs poorly for

complex flows involving severe pressure gradient, separation, strong streamline

curvature. The most disturbing weakness is lack of sensitivity to adverse pressure

gradients; another shortcoming is numerical stiffness when equations are integrated

through the viscous sublayer which are treated with damping functions that have stability

issues

Pros: Robust. Widely used despite the known limitations of the model. Easy to

implement. Computationally cheap. Valid for fully turbulent flows only. Suitable for initial

iterations, initial screening of alternative designs, and parametric studies (Menter 1993).

Cons: Performs poorly for complex flows involving severe pressure gradient, separation,

strong streamline curvature. Most disturbing weakness is lack of sensitivity to adverse

pressure gradients; another shortcoming is numerical stiffness when equations are

integrated through the viscous sublayer which are treated with damping functions that

have stability issues (F.R.Menter, Two-Equation Eddy-Viscosity Turbulence Models for

Engineering Applications 1994).

3.3.2 Standard K-Omega

A two-transport-equation model solving for kinetic energy k and turbulent frequency ω.

This is the default k–ω model. This model allows for a more accurate near wall treatment

with an automatic switch from a wall function to a low-Reynolds number formulation

based on grid spacing. Demonstrates superior performance for wall-bounded and low

Reynolds number flows. Shows potential for predicting transition. Options account for

transitional, free shear, and compressible flows. The k-e model uses the gradient

diffusion hypothesis to relate the Reynolds stresses to the mean velocity gradients and

the turbulent viscosity. Solves one equation for turbulent kinetic energy k and a second

equation for the specific turbulent dissipation rate (or turbulent frequency) w. This model

performs significantly better under adverse pressure gradient conditions. The model does

not employ damping functions and has straightforward Dirichlet boundary conditions,

which leads to significant advantages in numerical stability. This model underpredicts the

amount of separation for severe adverse pressure gradient flows.

Pros: Superior performance for wall-bounded boundary layer, free shear, and low

Reynolds number flows. Suitable for complex boundary layer flows under adverse

pressure gradient and separation (external aerodynamics and turbomachinery). Can be

used for transitional flows (though tends to predict early transition).


Cons: Separation is typically predicted to be excessive and early. Requires mesh

resolution near the wall.

3.3.3 SST K-Omega

Shear Stress Transport (SST) is a variant of the standard k–ω model. Combines the

original Wilcox k-w model for use near walls and the standard k–ε model away from walls

using a blending function, and the eddy viscosity formulation is modified to account for

the transport effects of the principle turbulent shear stress.Also limits turbulent viscosity

to guarantee that τT~k. The transition and shearing options are borrowed from standard

k–ω. No option to include compressibility.

Pros: Offers similar benefits as standard k–ω. The SST model accounts for the transport

of turbulent shear stress and gives highly accurate predictions of the onset and the

amount of flow separation under adverse pressure gradients. SST is recommended for

high accuracy boundary layer simulations.

Cons: Dependency on wall distance makes this less suitable for free shear flows

compared to standard k-w. Requires mesh resolution near the wall.

A Reynolds Stress model may be more appropriate for flows with sudden changes in

strain rate or rotating flows while the SST model may be more appropriate for separated

flows.

3.3.4 RSM

RSM(Reynolds-stress model): Use the same idea of flucturated flow but doesn’t accept

the assumption of isotropic flow. The turbulence stresses are modelled directly. Hence, 6

aditional unknowns are solved by 6 additional transport equations.

1. Better performance in complex flow regimes, especially in strongly curved streams

2. Improved stability as the asymptotic solution attracts all initial conditions (Charles G

Speziale 1991).

The main drawback of a Reynolds stress model is the introduction of six additional

transport equations for the turbulence stress terms. In addition, convergence problems

are identified when using the Reynolds Stress Model by Speziale et al. (1991).


4 CFD SIMULATION

CFD simulations are performed by a commericial numerical software STAR-CCM+.

The physics of the sloshing problem are considered in order to identify the key modelling

aspects. The correct application of CFD and how it can be used to model sloshing is

considered.

A space and time discretisation independence study is carried out to ascertain the

applicability of the results. Subsequently, the effect of including either a K- Epsilon or K -

Omega turbulence model as opposed to forcing laminar flow is examined.

4.1 Computational model The Free surface tank model discussed in this paper is scaled from a full size tank on an

OSV (Offshore supply vessel). Table below gives the main dimensions of the vessel.

Table 2. Main dimensions of the Vessel

Parameter Value [m]

Lpp 75.5

Breadth 21

Draught 6.8

Vertical center of gravity,VCG 7.6

Generally speaking, the liquid inside free surface tank should have a weight which

accounts to approximately 2-4% of the total ship displacement.This affect the tank

dimensions to a certain extent.

The tank breath is designed as large as possible in order to provide more damping

moment. Based on industrial experience, it is recommended that the designed tank

length should follow:

𝐿 = 3 + 3% × 𝐿𝑝𝑝 (20)

This is due to the ship hull structure factor. If the tank is designed too long, it may

occupy two bulkheads or even more which will affect the integrity of the ship hull.

Therefore, a recommended anti-roll tank for this particular OSV has following

dimensions:

Table 3. Full scale tank dimensions

Parameter Value [m]

Tank width 20

Tank length 5

Tank height 3

Vertical position of tank bottom 7.6

The full size tank is scaled to a model size tank by a scale factor 1:20. The model size

tank is therefore used in both numerical calculations and model tests.


4.1.1 Geometry description

Figure 17: Definition of geometry and tank dimensions: Free surface tank

Table 4. Model scale tank dimensions

Dimension Value [mm]

Tank width b 1000

Tank length L 250

Tank height H 150

Water level h 50, 75, 100

Vertical position of tank

bottom from rotation center

s 0

Fluid density 997.561 [kg/m3]

Dynamic Viscosity 8.8871E-4[Pa-s]

Table below shows the relation between water level in the tank model and the theoretical

natural period. Table 5. Natural periods of differnt filling tanks

Filling percentage Water level h[cm] Natural period [s] Natural frequency

33.3% 5.0 2.86 0.35

50% 7.5 2.33 0.43

66.6% 10.0 2.02 0.50

4.1.2 Tank motion

The tank motion is defined as a 1-DOF sinusoidal motion with the rotation center in the

middle of the tank bottom.

This paper will discuss three different rotation amplitudes: 3deg, 4.5deg, 6deg. And the

rotation periods includes from 1 second to maximum 10 seconds.

Motion frequency:

𝜔 =2π

T (21)

Motion displacement:


𝑥 = 𝐴 ∙ 𝑠𝑖𝑛(𝜔𝑡) (22)

Angular velocity:

�� = 𝐴 ∙ 𝜔 ∙ 𝑐𝑜𝑠(𝜔𝑡) (23)

4.2 Grid (Mesh) A grid (or mesh, the terms are often used interchangeably) is used to represent the

problem in computational fluid dynamics usually as a set of finite volume elements. The

grid design process is centered around the following trade-offs.

1. Spacing. The grid needs to be sufficiently fine so as to sustain conservation of mass

and momentum at an acceptable level. However, reducing grid size will increase the

computational cost and memory requirements. The rate of increase depends on the

type of solution algorithm used.

2. Resolution. Grid spacing needs to be sufficiently small to resolve the flow in all

regions of the computational domain. This is especially important when using a

turbulence model, where the position of the first near-wall grid point can have a

significant influence on the CFD output quality. This is discussed by (Wilcox 1998) in

considerable detail. When the grid is too coarse, local flow features are smeared and,

especially when considering a sloshing flow, pressure spikes are not resolved with

sufficient detail.

3. Geometry. Although not an issue in the current study, the grid must provide a

sufficiently accurate representation of the geometry used. This becomes very

important when there are small changes (e.g. ripples) on a surface, or a body has

particular details influencing the flow.

As the grid represents the problem in computational space, one should always ascertain

that any CFD result is independent of the grid used. This has been considered by the

Maritime CFD best practice guidelines (Consultants 2003).

In this report, the tank geometry is meshed by Trimmer model using a base size 0.005m.

In order to get better results, volumetric control in Z direction is introduced. The relative

Z size is 0.0025m which is 50% of the other two directions.

Figure 18: Tank mesh


4.3 Temporal discretization scheme Star CCM+ offers two different time discretisation schemes, a first order and second

order backward Euler scheme. The first order scheme approximates the time derivatives

in integral form,

(24)

While this scheme is robust, it does induce numerical diffusion. Therefore, a second order

approximation of the time derivative, given as

(25)

And

The second order discretisation is conservative. However it may give nonphysical

solutions when the flow experiences severe changes in the time domain. An advantage of

the second order model is the extrapolation of an initial guess for the current timestep i

based on previous timesteps. As fewer iterations are required to determine the new

solution, computational times are significantly reduced. The current investigation will

assess the difference between the two time marching schemes.


Figure 19. Temporal discretization scheme comparison 1

Figure 20. Temporal discretization scheme comparison 2

From the comparisons above, it can be easily found that there are great differences

between the 1st-order and 2nd-order schemes when simulating a severe sloshing. Moment

in Figure 20 shows some differences as well. It is difficult to make the decision to choose

which. In this case, model tests are needed as a reference. Further discussion can be

found in Model test part.

4.4 Time step control As the velocity of the flow varies throughout the simulation, it is useful to adjust the

timestep according to the flow velocity. This is governed using the Courant number,

which is the rate of flow speed with which numerical disturbances propagate. The

Courant number at node i is defined by (Hirsch 1988) for finite volume CFD as

𝐶𝑁 = 𝑢𝑖∆𝑡

∆𝑥𝑖 (26)

where ui is the flow velocity, t the time step and xi the grid spacing at node i. As flow

velocity varies throughout the flow field, so will the Courant number.

Typically, it is recommended that 𝐶𝑁 ≤ 1.0 to maintain a stable solution if the flow field is

not known a priori (John D Anderson 1995). If the grid spacing ∆𝑥 were to be halved, the

Courant number may be kept constant only by halving the time step as well. This

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0

10

20

30

40

0 5 10 15 20 25 30

Mo

men

t [N

m]

Time [s]2nd order 1st order


illustrates the interaction between grid size and time discretisation, underlining the

importance of considering the time step size when generating the grid.

Systematic time step variations for sloshing flows have been carried out by Rhee (2005)

uses a timestep of 0.001 sec. In order to find a suitable time step for the case, here we

investigate three different time steps and make a comparison.

Here we compare three different time steps 0.005s, 0.001s and 0.0005s. Tank filling

percentage is 33%; excitation amplitude is 4.5 degree; mesh size is 0.005m as

predefined. Figure 21 shows the maximum courant number of each time step during a

complete simulation.

Figure 21. Courant Number comparison


Table 6 summerize the courant numer of three simulations. Figure 22 illustrate the

damping moment of each case. We can find significant increase on moment amplitude

from time step 0.005s to time step 0.001s. While there is not such a big moment

difference between time step 0.001s to time step 0.0005s. Both moment amplitude and

phase are stable, which means that the result is converged. Therefore, it is sufficient to

sue the criterion which courant number is 1.0.

Table 6. Time steps and Courant Number

Time step CFL

0.005 < 3

0.001 ≈ 1

0.0005 ≈ 0.5

Figure 22. Time steps comparison

4.5 Turbulence modelling The successful numerical solution depends on sufficient grid resolution to capture all flow

features. Turbulence takes place at often very small spatial and time scales. In order to

capture the effects of turbulence using the governing equations in their present form,

grids with extremely high resolution and very small time steps would be required.

In this paper, two turbulence models (K-Epsilon and K-Omega) are compared and

discussed. As explained in chapter 3.3, K-Epsilon turbulence model is a two-equation model in which transport equations are solved for the turbulent kinetic energy 𝑘 and its

dissipation rate 𝜀 . While K-Omega turbulence model uses turbulent frequency ω instead

of dissipation rate 𝜀.

Basically, a K- Epsilon model is robust, and widely used despite the known limitations of

the model. Compared to other turbulence models, it is easy to implement and

computationally cheap. Besides, it is suitable for initial iterations, initial screening of

alternative designs, and parametric studies.

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30

40

0 5 10 15 20 25 30 35 40 45

Mo

men

t [N

m]

Time [s]

Time step comparison

Time step 0.005 Time step 0.001 Time step 0.0005


K-Omega model has superior performance for wall-bounded boundary layer, free shear,

and low Reynolds number flows. It is suitable for complex boundary layer flows under

adverse pressure gradient and separation (external aerodynamics and turbomachinery).

It also can be used for transitional flows (though tends to predict early transition).

Figures below compare these two models in greater details. Here we compare three

different excitation periods 4s, 3s and 2s.

Figure 23. Turbulence modelling at 4s

When the excitation period is 4s, there is nearly no difference between K-Epsilon and K-

Omega models on the moment curves. However, if we look carefully at the liquid inside

the tank, we may say that K-Omega provides more excessive details in water-air

interface and interaction between water and gas as shown in Figure 23.

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20

25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Mo

men

t [N

-m]

Time [s]

3D, 33% filling percenatge, Roll amplitude 3deg, Excitation period 4s

K-Epsilon K-Omega



If we reduce the excitation period to 3s, we may find a slight moment defference

between K-Epsilon and K-Omega models. Therefore, we pick up two moments 8.78s and

13.28s when the difference is maximum. From the figures above, it can be found that the

difference mostly due to the accuracy of predictions from two models. Similar to 2s, K-

Omega provides more excessive details in water-air interface and interaction between

water and gas in this case. This results in slightly less damping moment.

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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Time [s]


K-Epsilon K-Omega



If we keep on reducing the excitation period to 2s, obvious moment difference can be

found in the curve. We pick up the moment 15.38s when obvious difference occurs.

Different from previous cases, this time the difference occurs when the liquid hit the side

wall. At the same time, turbulence flow and liquid seperation appears close to the side

wall . All of these result in a less damping moment for K-Omega model.

In all the three cases, different turbulence models may affect the amplitude of damping

moment to some extent, while it doesn’t affect the moment phase at all. One thing

should also be noticed is that in our cases, the liquid inside the tank is water and we

neglect the compressibility of liquid and gas. If one simulate for instance LNG and

consider the compressibility, different turbulence models should be carfully treated and

fully discussed.

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Mo

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t [N

-m]

Time [s]


K-Epsilon Moment K-Omega


4.6 2D vs 3D model Generally speaking, turbulence is a three-dimensional time-dependent phenomenon,

therefore the simulations should use 3D model. However, a 2D model is computionally

cheap compared to a 3D model when using same dimensions. Therefore, it is of interest

to investigate the differences beteen 2D and 3D simulations.

The grids/meshes of 2D and 3D model are compared in the following table. Notice that

the calculating environment is Windows 8.1 x64, Intel(R) Core(TM) i7-3635QM CPU @

2.40GHz 2.40GHz, 8G RAM.

Table 7. 2D and 3D comparison

cells Interior Faces Vertices Computional time

2D 12000 23740 12261 20 mins for 10 s

3D 127296 376096 171609 233 mins for 10s

Ratio 1:10.6 1:15.8 1:14 12:1

Next, we will compare 2D and 3D simulations when excitation period is 2s, 2.5s, 3s, 4s,

5s, respectively. In order to make sure that 2D simulations are accurate enough, we use

a smaller time step 0.001s.

When the excitation period is relatively small for instance 2.5s as shown in Figure 26,

some differences appears between 2D and 3D simulations. An obvious reason is that 2D

simulation neglect the vortex in tank length direction. Besides, it does not include the

effect of the tank side walls. However, these two aspects may affect the result only when

the flow inside the tank is fully turbulent. When we increase the excitation period to 3s,

4s and 5s, it can be found that the differences between 2D and 3D simulations become

smaller and smaller.


Figure 26. 2D vs 3D at excitation period 2.5s

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2D vs 3D, Roll amplitude 3deg, Excitation period 2.5s

2D model 3D model

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2D vs 3D, Roll amplitude 3deg, Excitation period 2s

2D model 3D model

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2D model 3D model


Figure 27. 2D vs 3D at different excitation periods

Based on previous discussion, the moment differences between 2D and 3D simulations

are rather small. Therefore, we may accept 2D simulations only in our specific cases.

Though turbulence is a three-dimensional time-dependent phenomenon, we may use 2D

simulations to initial iterations, initial screening of alternative designs, and parametric

studies because it is easy to implement and computationally cheap.

However, when the excitation period is very small and the sloshing inside is severe, 3D

simulation is recommended and it is more reliable than a 2D simulation.

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2D model 3D model

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5 TANK PERFORMANCE DISCUSSION

5.1 Moment and phase This part will discuss how to get the moment and phase from the results we received

from simulations.

One can get the moment data after running a simulation in Star CCM+. As an initial

analysis, one need to select the time region when the system reach steady status. One

possible following step is to use Matlab Curving fitting tool to find out the expression of

this curve and gain the mathematical equation. Then one can get the moment amplitude

and phase lag at one excitation period. Repeat the procedure over and over again. The

moment curve and phase curve can be achieved.

Figure 28. Data post-processing

5.2 Natural period In chapter 2.4.2, we have already calculated the highest natural periods of the anti-roll

tank. Table 5 gives the highest natural periods of different filling tanks. But are they the

same with the realistic natural periods?

Since we already get the time domain results of the tank damping moment, we may

therefore use Fast Fourier Transform(FFT) to find out the frequency domain. This process

is implemented in 20-sim. Then the excitation frequency and natural frequency can be

found directly.

Here, we select 33% filling percentage and 3deg excitation amplitude as an example to

discuss.


Excitation period: 1s

Excitation period: 1.2s

















Figure 29. Time domain and frequency domain

As the excitation period varies from 1.0s to 1.75s, two dominated frequencies can be

found in the frequency domain figure. One equals to the excitation frequency, and the

other remains a stable value, which should be the natural frequency of the system. In

this case, the natural frequency of the tank is 0.33. Besides, comparing the peak values

of these two frequencies, we can find that the function/effect of the natural frequency

reduces with the increase of excitation period. This also implies that the system transfers

from stiffness dominated region to damping dominated region.

When the excitation period reaches 2.0s, only one dominated frequency (excitation

frequency) can be found in the domain. It indicates that the system is dominated by

external excitation and enters in the damping dominate region. When excitation period is

3.0s, which equals to the natural period, the peak of the frequency reaches the highest

value among all the excitation periods. This is where the resonance happens. Comparing

3.0s with adjacent periods, we can also find that the peak value of 2.5s and 4.0s are

close to the peak value of 3.0s. This phenomenon is so called frequency locking(or “lock-

in”).


If we increase the excitation period to 7.0s and even higher, the natural frequency

appears again in the frequency domain. This is because the excitation period is far from

the natural frequency. The system enters into inertial dominated region.

Figure 29 shows the distribution of these three regions. Basically, the designers wish

their ART located in the damping dominated region so that they can achieve better

performance. This is also why some people name it “damping tank”. In order to

maximize the performance, the designate tank is supposed to have a -90 deg phase

angle. In this case, the amplitude of the moment is approximately 20 Nm which is close

to the maximum amplitude.

Figure 30. Different dominated regions

Using similar idea, natural frequencies of the other two filling percentage are found as

shown in Table 8. It can be found that the natural frequencies from simulations are

slightly smaller than those calculated by formula.

Table 8. Natural frequency comparison

Filling

percentage

Water level

h[cm]

Highest natural

period

Natural

frequency

Natural frequency

from simulation

33.3% 5.0 2.86 0.35 0.33

50% 7.5 2.33 0.43 0.41

66.6% 10.0 2.02 0.50 0.49


5.3 Excitation Amplitude This chapter will investigate the performance of the FST at different excitation

amplitudes.

In figure 31, the moment is expressed as moment per unit roll to make the values

comparable with other amplitudes of roll. The total moment can be found by multiplying

with the amplitude of roll motion. The amplitude of the moment can also be read out of

the time series.

When multiplying each curve by its roll amplitude it is clear that the black curve

multiplied by six gives higher values than the blue curve multiplied by two. A higher roll

amplitude represented by the black curve gives therefore higher damping moments.

Another aspect can be seen from the graph: The distances between the single curves are

not equal. The damping moment increases not linear with the roll amplitude.

The lower graph in figure 32 shows the phase of the moment relative to the rig motion in

degrees over the roll periods. For the first period of four seconds and for the large

periods towards the end, there is nearly no phase difference because either the water is

too inert to follow the rig motions or it can easily follow the motions at large periods. The

phase of the moment can be read out of the recorded time series after the measured

forces are multiplied by the lever arm to the centre of gravity.

Figure 31. Moment comparison of different excitation amplitudes (Model scale)

Figure 32. Phase comparison of different excitation amplitudes (Model scale)

0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6 7

Mo

men

t [N

m/r

ad]

Roll period [s]

Comparison of different amplitudes of roll (Model scale)

3deg roll amplitude 4.5deg roll amplitude 6deg roll amplitude

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4 5 6 7

Ph

ase

of

mo

men

t [d

eg]

Roll period [s]


3deg roll ampliude 4.5deg roll ampliude 6deg roll ampliude


We simply transfer the CFD simulations from model-scale to full-scale by using following

equations:

Scale factor:

𝜆=20

Roll Period:

𝑇𝑓𝑢𝑙𝑙−𝑠𝑐𝑎𝑙𝑒 = √𝜆 ∙ 𝑇𝑚𝑜𝑑𝑒𝑙 (27)

Bending Moment:

𝑀𝑓𝑢𝑙𝑙−𝑠𝑐𝑎𝑙𝑒 = 𝜆4 ∙ 𝑀𝑚𝑜𝑑𝑒𝑙 (28)

Figure 33. Moment comparison of different excitation amplitudes (Full scale)

Figure 34. Phase comparison of different excitation amplitudes (Full scale)

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30

Mo

men

t [M

Nm

/rad

]

Roll period [s]

Comparison of different amplitudes of roll (full scale)


-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0 5 10 15 20 25 30

Ph

ase

of

mo

men

t [d

eg]

Roll period [s]

Comparison of different amplitudes of roll (Full scale)



5.4 Filling level/percentage The main dimensions of a ART are defined in design phase. Therfore, change the filling

percentage becomes the only way to ajust the natural frequency of the tank. In this part,

we will discuss three different filling levels: 5cm, 7.5cm and 10cm, corresponding to

33%, 50% and 66% filling percentage.

Figure 35. Moment comparison of different filling levels (Model scale)

Figure 36. Phase comparison of different filling levels (Model scale)

Normally, higher filling level means more liquid inside the tank which will provide more

static forece thus create more damping moment. This phenomenon can be found from 1s

to 2.5s in Figure 35.

The natural period of the tank reduces as the filling level increases. It can be found that

higher filling level create higher peak moment.

When the sytem enters in inertial dominated region, the tank with more liquid generates

less moment. More liquid inside the tank means more mass, which creates more inertial

impact. Therefore, it is more difficult to excite the liquid motion. Besides, the system with

more mass enters in inertial dominated region much more early than those with less

mass. This can also be found in the phase curve.

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7

Mo

men

t [N

m/r

ad]

Roll period [s]


5cm filling height 7.5cm filling height 10cm filling height

-200

-150

-100

-50

0

0 1 2 3 4 5 6 7

Ph

ase

of

mo

men

t [d

eg]

Roll period [s]




When the phase close to 0 (after 3.0s), the system enters in interial dominated region.

Besides, it indicates the resonance occurs when the phase angle close to -90deg.

Then we transfer the model scale result to full scale result as shown in Figure 36.

Considering the resonance period of the vessel is around 13s (Figure 37), we can found

the tank performance at 50% filling is much better that it with 33% filling. The optimal

filling level for this particular tank is between 50% and 66%.

Figure 37. Moment comparison of different filling levels (Full scale)

Figure 38. Roll RAO of the OSV

5.5 Cover Panel Effect This part will discuss the influence of the cover panel. The predefined tank height is

15cm. In order to make a comparion, a second tank with a doubled height 30cm is

introduced. Figure 39 show the snap shots at different moment.

0

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Mo

men

t [M

Nm

/rad

]

Roll period [s]



0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

RA

O

RO

LL

η(4

)/A

[deg/m

]

WAVE PERIOD [sec]

DISPLACEMENTS

Project: Base Case - Operability

Base case ; 0.00kn 90.0°



Figure 39. Snapshots of cover effect

From the figures above, we can find that the cover panel has influence when the water

hits side walls which will result in a less damping moment. Futhermore, the water drops

falling down from the cover panel affect the flow below. This results in the phase

different as shown in solution time 5.84s, 6.94s.

As time goes on, the cover effect increase more. The wave pattern inside the tank is

different. At time 10.88s and 20.3s, we can see that the wave in the limited height tank

becomes a combined wave instead of hydraulic jump and travalling wave.

Figure 40 gives the damping moment of these two tanks. It is suggested that using a

finite tank height to do pure sloshing study is better since it may negelect the cover

panel effect and the system can reach steady state in a shorter time.

Figure 40. Cover effect

-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40

Mo

men

t [N

m]

Time [s]

Cover effect

No Cover Cover


6 EXPERIMENT AND VALIDATION

Numerical method provides complete analysis and solutions to the ART problem.

However, it is difficult to convince the accuracy of the results. It is therefore necessary to

do validation by using experiment data. Two ways of test validation are provided in this

thesis. One validation comes from ShipX database, which is established by MARINTEK in

2014. The other is the model test performed by the author at Aalesund University

College. Next will give an introduction of these validations.

6.1 Experimental data from ShipX

6.1.1 Tank design in ShipX

MARINTEK uses the ShipX Vessel Responses (VERES) programme to include the effect of

motion control systems on the ship motions. Different types of tanks can be handled.

Either model test data from a given tank or data from systematically performed tests

stored in a database can be taken as input for designing a tank using the VERES

programme.

The user manual written by Fathi (2014) (Fathi 2014) gives a guidance on how a rough

design of a free surface roll damping tank is determined according to the sea

performance of the actual ship. First, the ship motion transfer function is found in VERES

which is defined by the given vessel geometry and the loading condition. Mass and

position of the tank are already included but the free surface effects are of course not

incorporated because the original ship performance shall be treated. It should be noticed

that the sea state where the ship shall operate is an important factor in the transfer

function, as larger waves will lead to larger roll amplitudes of the vessel. By plotting the

transfer function one can find the natural roll period of the ship.

Second, tank geometry and maximum filling height are specified. Additional options as

damping grids can be chosen. The programme then calculates which filling level is

optimal for each roll period. An important assumption is made in this programme: It is

assumed that the roll damping increases linear with increasing roll amplitude, also at

resonance.

Third, by knowing now the natural roll period of the ship the optimal filling level can be

read out of a plot from the second step. Finally, the modified ship transfer function with

included roll reduction can be plotted. If the achieved damping should not be sufficient, a

longer tank as well as a higher location of the tank would increase the effect of the roll

damping moment. Sometimes, a second tank is installed. Such operating vessels have

often bilge keels in addition as initial configuration. Some of these ships cannot operate

with the tank when the GM-requirements are not fulfilled. Bilge keels need to be included

in the ship motion calculations.

6.1.2 Tank database

The data in the VERES database are collected from literature, as well as systematic

model tests in MARINTEK’s roll simulator. In the model tests, a roll amplitude of 0.07

radians (4 deg) is applied in the roll simulator.

Tank filling:

0.04 ≤ ℎ/𝑏 ≤ 0.10

Vertical positions:

−0.2 ≤ 𝑠/𝑏 ≤ 0.40


6.1.3 Data comparison

The model in ShipX is a full-scale OSV which defined in chapter 4.1. So the tank installed

on this vessel is full scale as well. Therefore, we will compare the data from CFD

simulations and experimental data from ShipX in full scale size.

Figure 41. Comparison between simulations and database in ShipX

The moment plotted in Figure40 is in MNm/radians in order to make the values

comparable with other amplitudes of roll. It should be noticed that the model tests in

ShipX only provides a roll amplitude of 0.007 radians (4deg). While the CFD simulations


have three different roll amplitudes: 3deg, 4.5deg and 6deg. The trend of the curves

maintain consistency especially in damping and inertial dominated regions. In high

frequency (small excitation periods) region, the results do not match perfectly.

Further comparisons are performed in following figures. Moment and phase at different

filling levels are compared separately.

Figure 42. Moment comparison at different filling percentages between simulations and database in

ShipX


Figure 43. Phase comparison at different filling percentages between simulations and database in

ShipX

All comparisons give similar result. The trend of the moment curves maintain consistency

especially in damping and inertial dominated regions.

While the phase curves from the experiments and simulations maintain consistency in

inertial dominated region. In the damping dominated region, the phases from simulations

are normally larger than those from experiments. Besides, it shows some difference in

high frequency (small excitation periods) region.


6.2 Model Test set up and plan Model tests are designed and performed at Aalesund University College, Aalesund,

Norway. This chapter will introduce the overall design of the tank platform and test

plans.

6.2.1 Model Tank Design

Figure 44. Tank design

The retangular tank was made of plexiglass with the same dimensions as in the CFD

simulations. The thickness of the tank side wall is 6mm and 10mm for bottom. An

additional cover is needed to make sure that the liquid inside won’t splash out. Detailed

drawing can be found in Appendix A.

6.2.2 Platform Set Up

The designed tank platform contains three parts: foundation structure, tank carrier and

link-motor part as shown in Figure 45. Details can be found in the Appendix 3D models.

Figure 45. Platform design and physical model

Physical model was manufactured at Aalesund University College as shown in Figure 46.


Figure 46. Model Platform overview

Table 9. Main conponents on the phatform

Component Description

Motor AC-servo JVL MAC800 X1;

Driver: Mactalk

Gear box Gear ratio 1:50

Sensors Full bridge strain sensor X1

Half bridge angle sensor X1

Data logging system Hardware: Spider 8;

Software: Catman

Camera CS-Mount 6.2mm f/1.8 Non-Distortion Fixed Lens X2

Driver: StCamSWare

Arrangement: One fixed camera; the other moving with the tank.

Mirror 1.2m*0.4m

Table 9 lists the main component installed on the platform. The motor used in the tests is

an AC-servo motor: JVL MAC800, which controlled by integrated software Mactalk. The

uniform rotation of the motor is then transferred to a roll motion of the tank carrier by

using a crank link. It is essential to make sure that the motion of the tank carrier is the

correct motion we expected. The whole platform has been simulated in NX with a

constant motor speed. A snap shot can be found in Figure 47. The motion of the point is

plotted in Figure 48.


Figure 47. Simulation view in NX

Figure 48. Tank motion result from simulation

Figure 49. Tank motion error from simulation

(100)

0

100

200

300

400

500

600

700

800

900

0 5 10 15 20 25 30 35

Time

Displacement: X, Y, MAG [mm]

J003_X,Displacement(abs)J003_Y,Displacement(abs)J003_MAG,Displacement(abs)

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0 5 10 15 20 25 30 35

Dis

pla

cem

ent

[mm

]

time[s]

Error [mm]


In Figure 49, the difference between the displacement in Y direction and magnitude

displacement is less than 0.6mm. Compared to the magnitude displacement, the error

only account for 1% which is acceptable in our case.

The platform is designed to have the capacity to test different types of tanks including

FST, U shape tank, C shape tank and cylinder tank. The maximum tank size should be

less than 0.7m in length and 1.5m in breadth.

Table 10. Phatform test capacity

Maximum Tank size Tank length L ≤ 0.7m

Rotation center 0cm; ±10cm; ±20cm; 30cm

Excitation amplitude 3deg; 4.5deg; 6deg; 7.5deg; 9deg

Excitation period Minimum 1s

6.2.3 Test Preparation and Plans

Before doing the experiment, the strain transducers on the platform should be calibrated

first. This is carried out by measuring their output while applying a certain force on them.

The angle transducers are calibrated by measuring the highest and lowest point. After

finishing each test, the platform and the transducers are requied to be set at zero

position for doing the following tests.

Due to some reasons, currently the platform can only test the anplitude of 6 degree.

Therefore, in this paper we would perform the tests with the excitation amplitude of 6

degree. Detaled text plan can be found in table 11. Notice that the tests without water

are also required at each excitation periods.

Table 11. Test plan

Test

number

Amplitude Excitation

Period [s]

Motor Input With/Without

liquid

001 6 deg 1 3000 both

002 6 deg 1.25 2400 both

003 6 deg 1.5 2000 both

004 6 deg 1.75 1714 both

005 6 deg 1.8 1667 both

006 6 deg 1.9 1579 both

007 6 deg 2 1500 both

008 6 deg 2.5 1200 both

009 6 deg 3 1000 both

010 6 deg 4 750 both

011 6 deg 5 600 both

012 6 deg 6 500 both

One thing should be noticed is that the Motor input is the value where we put in the

Mactalk software. In this case, we have already transferred the excitation period to the

motor rotation speed and mutiplied by the gear box ratio 50.


6.2.4 Data post-processing

After finishing all the tests, we can get the moment data both from the tests without

liquid and the tests with liquid. The basic idea is to substract the moment without liquid

from the moment with liquid. In this way, the liquid moment can be achieved. Then we

can do the similar procedure to get the moment amplitude and phase lag from these

data.


7 DISCUSSION

7.1 Comparison between model tests and simulations This part compares the data from simulation with those from model tests. Discussions of

model tests and simulations will be conducted here.

The following series of snap shots give a good comparisn between the model test and

numerical simulation when the excitation period is 6s.



Figure 50. Comparison between model test and simulation


It can be found that the simulation has shown extremly good consistancy with the model

test at excitation period of 6s. Figure 51 and 52 illustrate the moment and phase

comparisons.

Figure 51. Moment comparison between model test and simulation

Figure 52. Phase comparison between model test and simulation

The simulations remain stable in stiffness dominated (1s-1.5s) and inertial dominated

(3s-6s) regions. Both moment and phase have shown high consistency in these regions.

The differences of the moment is less than 10% and the phase has a difference less than

6%.

However, in the damping dominated region (1.5s-3s), the difference between model test

and simulations should not be overlooked especially when the excitation period close to

the natural period of the tank where resonance occurs.

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7

Mo

men

t [N

m]

Roll period [s]

Moment comparion [Nm]

Experiment CFD

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0 1 2 3 4 5 6 7

Ph

ase

[deg

]

Roll period [s]

Phase comparion [deg]

Experiment CFD


One of the reasons is the high courant number in the simulation. When the excitation

amplitude is 6deg, the motion and velocity is twice as much as that at 3deg amplitude.

Therefore, smaller time step is required to keep the courant numer at an acceptable

level. Therefore, the results will be more realistic and reliable. Another possible solution

to imporve the simulation result is to use 2nd-order temporal discretization scheme

instead of 1st-order scheme. Then we can keep the numerical diffusion at a lower level.

7.2 Challenges in the thesis project CFD has been an useful but challenging tool in applications. One should be very careful

when set up the simulation models. It is easy to run a simulation but challenging to make

sure that the simulation is accurate. Therefore, it is nesseary to establish a strong

theoretical background to fully understand the algorithm and settings behind the

software.

Model platform design and establishment took more time than expected. I started this

thesis project from early February if not considering the previous literature study time. It

took me approximately a month to design the model platform and make evaluation, two

and half month to manufacture and install the motor, tank, camera and other

components. Acturally, there was only one week for me to do the model tests which

made me anxiety. Because if the simulation results were far away from the model tests,

there was no time left for me to make modifications or redo the simulations. Therefore, I

have to force myself to think as much as possible at the very beginning to make sure

everything goes in the right way. Fortunately, the results are very positive both for the

simulations and the model tests.


8 CONCLUSIONS

In this thesis project, the development of anti-tanks is summarized as well as the

numerical methods. Different types of anti-roll tanks are introduced and compared.

One contribution of this project is developing guidelines for anti-roll tank simulation. A

numerical model is introduced and discussed. And this particular model is suitable for a

general thermostatic incompressible sloshing problem.

The main part of this project is studying the anti-roll tank performance. Firstly, the

natural period of the partially filled tank is investigated and discussed.

Then different excitation amplitudes have been investigated. It has been found that roll

damping increases nonlinear with increasing roll amplitude at resonance frequency

region. Smaller excitation amplitude will result in higher amping moment. However, the

damping moment has liner increase when the excitation period away from the damping

dominated region.

Besides, different filling levels are studied and discussed. It found that the roll damping

increases with increasing liquid level and the damping dominated region turned to be

more narrow. If known the natural period of vessel roll motion, the optimal filling level

can therefore be found.

Furthermore, it compares the results from simulations with the exsiting experimental

database. The comparions have shown good consistency. The model tests, which

designed and performed by the author, have given a strong validation for the numerical



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Veritas. Oslo, Norway, 1974.

Faltinsen, O.M., Timokha, A.N. Sloshing. 2009.

Fathi, Dariusz Eirik. ShipX Vessel Responses (VERES), Motion Control Extension.

MARINTEK A/S, 2014.

Frahm, H. "Results of trails of the anti-rolling tanks at sea." Transactions of Institution of

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Goudarzi, M. A. and Sabbagh-Yazdi, S. R. "Investigation of Nonlinear Sloshing." Soil

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McGraw Hill, Inc., 1995.

Lee, B.S., Vassalos, D. "An investigation into the stabilization effects of anti-roll tanks

with flow obstructions." International Shipbuilding Progress. 1996. 43(433),70.


Lewis, E. V. "Principles of Naval Architecture, Volume iii, Motions in Waves and

Controllability." Society of Naval Architects and Marine Enogneers, 1989.

Liu, Z., Huang, Y. "A new method for large amplitude sloshing problems." Journal of

Sound and Vibration, 1994: 185-195.

Martin, J.P. "Roll Stabilisation of Small Ships." Marine Technology, 1994: Vol. 31, No. 4.

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Fluid Dynamics Conference. 1993. 93-2906.

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Sames, P.C., Marcouly, D., Schellin, T.E. "Sloshing in rectangular and cylindrical tanks."

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Solaas, F. "Analytical and Numerical Studies of Sloshing in Tanks." NTH. Trondheim,

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411.

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APPENDIX

Appendix A Tank Drawings

Appendix B 3D model of Tank Platform (digital)

Appendix C Test data files (digital)

Appendix D Simulation files (digital)

Appendix E Animations of comparions (digital)

Appendix F Draft paper version of the thesis work

1 / 9

（Initial draft）

Computational Fluid Dynamics (CFD) study on Free surface anti-roll tank and experimental validation

ABSTRACT


personnel. Anti-roll tanks are tanks fitted onto ships in order to improve their response to

roll motion, which is typically the largest amplitude of all the degrees of freedom. In this

paper, a numerical model based on Volume of Fluid (VOF) is introduced to simulate the

liquid sloshing inside a free surface tank (FST). Through the simulation of the model, the

performance of the FST is fully investigated when the tank is excited by different periods

and different amplitudes. Besides, different tank filling levels are studied to find out the

optimal filling level. The results are proved by the existing experimental database.

Furthermore, results from model test gives further validation to the numerical model. The

comparisons between the model tests and simulations have shown good consistency.

Further improvements are proposed in the end.

Key words: Volume of Fluid (VOF), free surface tank (FST), model tests, validation

INTRODUCTION

The subject of roll stabilisation has been investigated for many years by Naval Architects

and designers. The main reason for this continued interest is that for roll, unlike pitch or

other degrees of freedom, the forces required to create a stabilising effect are relatively

small.

There are a great number of roll stabilisation devices available for use on vessels today.

Many of these were initially developed for use on large ships but are increasingly being

installed on smaller vessels where the subsequent benefit to the vessel’s motions is often

more apparent. This is due to the relatively high transverse stability, and short natural

roll periods of smaller vessels (Martin 1994). The prevalence of shorter waves also

means that smaller vessels are more likely to encounter resonant wave conditions and

thus violent rolling. Therefore, this paper invetigated the tank performance on a offshore

supply vessel (OSV) byusing CFD tool and model tests.

NUMERICAL METHOD

The motion of a fluid can be described by a set of partial differential equations expressing

conservation of mass, momentum and energy per unit volume of the fluid. The Navier

Stokes equations for three dimensional incompressible fluid flow can be written in

conservation form as follows

(1)

(2)

where

• 𝑣 is the velocity field vector;

• p is the pressure;

2 / 9

• ρ is the mass density;

• ν is the kinematic viscosity;

• g is the acceleration of gravity;

• 𝑒3 is the unit vector pointing upwards;

• ∇ is the gradient operator;

• ∇⋅ is the divergence operator;

• ∇⋅∇ is the Laplace-operator;

The interface between the phases of the mixture is resolved by using Volume of Fluid

(VOF) model.

TANK DESCRIPTION

A series of tests with free surface anti-roll tank in 1DOF harmonic motion was conducted

at Aalesund University College. The results of these experiments in terms of moment

amplitude and phase angle were used to validated the simulation model.

Table 1. Main dimensions of the Vessel

Parameter Value [m]

Lpp 75.5

Breadth 21

Draught 6.8

Vertical center of gravity,VCG 7.6

Table 2. Full scale tank dimensions

Parameter Notation Full scale Model scale

Value [m] Value [mm]

Tank width b 20 1000

Tank length L 5 250

Tank height H 3 150

Filling level h 1;1.5;2 50;75;100

Rotation point above the tank bottom

s 0 0

Figure 1: Definition of geometry and tank dimensions

3 / 9

Normally, the liquid inside free surface tank should have a weight which accounts to

approximately 2-4% of the total ship displacement. The tank breath is designed as large

as possible in order to provide more damping moment. Based on industrial experience, it

is recommended that the designed tank length should follow:

𝐿 = 3 + 3% × 𝐿𝑝𝑝

The full size tank is scaled to a model size tank by a scale factor 1:20. The model size

tank is therefore used in both numerical calculations and model tests. The tank motion is

defined as a 1-DOF sinusoidal motion with the rotation center in the middle of the tank

bottom.

MODELLING

In this paper, the tank geometry is meshed by Trimmer model using a base size 0.005m.

In order to get better results, volumetric control in Z direction is introduced. The relative

Z size is 0.0025m which is 50% of the other two directions.

Star CCM+ offers two different time discretisation schemes, a first order and second

order backward Euler scheme. Both of them are tested and recommendation will be

given. A time step of 0.001s is applied in the numerical simulation. Thus, the Courant

number, which is the rate of flow speed with which numerical disturbances propagate,

will be controlled below 1.0. The flow inside is simulated as turbulent flow. K-Epsilon

turbulence model is applied here.

2D FLOW VS 3D FLOW

Generally speaking, turbulence is a three-dimensional time-dependent phenomenon,

therefore the simulations should use 3D model. However, a 2D model is computionally

cheap compared to a 3D model when using same dimensions. Therefore, it is of interest

to investigate the differences beteen 2D and 3D simulations.

The grids/meshes of 2D and 3D model are compared in the following table. Notice that

the calculating environment is Windows 8.1 x64, Intel(R) Core(TM) i7-3635QM CPU @

2.40GHz 2.40GHz, 8G RAM.

Table 7. 2D and 3D comparison

cells Interior Faces Vertices Computional time

2D 12000 23740 12261 20 mins for 10 s

3D 127296 376096 171609 233 mins for 10s

Ratio 1:10.6 1:15.8 1:14 12:1

Next, we will compare 2D and 3D simulations when excitation period is 2s, 2.5s, 3s, 4s,

5s, respectively. In order to make sure that 2D simulations are accurate enough, we use

a smaller time step 0.001s.

When the excitation period is relatively small for instance 2.5s as shown in Figure 2,

some differences appears between 2D and 3D simulations. An obvious reason is that 2D

simulation neglect the vortex in tank length direction. Besides, it does not include the

effect of the tank side walls. However, these two aspects may affect the result only when

the flow inside the tank is fully turbulent. When we increase the excitation period to 3s,

4s and 5s, it can be found that the differences between 2D and 3D simulations become

smaller and smaller.

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Figure 2: Comparison between 2D and 3D flow

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0 5 10 15 20 25

Mo

men

t [N

m]

Time [s]

2D vs 3D, Roll amplitude 3deg, Excitation period 2.5s

2D model 3D model

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t [N

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2D model 3D model

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2D model 3D model

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2D model 3D model

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Based on previous discussion, the moment differences between 2D and 3D simulations

are rather small. Therefore, we may accept 2D simulations only in our specific cases.

Though turbulence is a three-dimensional time-dependent phenomenon, we may use 2D

simulations to initial iterations, initial screening of alternative designs, and parametric

studies because it is easy to implement and computationally cheap.

However, when the excitation period is very small and the sloshing inside is severe, 3D

simulation is recommended and it is more reliable than a 2D simulation.

EXCITATION AMPLITUDES

Figure 3. Moment comparison of different excitation amplitudes (Model scale)

Figure 4. Phase comparison of different excitation amplitudes (Model scale)

In figure 3, the moment is expressed as moment per unit roll to make the values

comparable with other amplitudes of roll. The total moment can be found by multiplying

with the amplitude of roll motion. The amplitude of the moment can also be read out of

the time series.

When multiplying each curve by its roll amplitude it is clear that the black curve

multiplied by six gives higher values than the blue curve multiplied by two. A higher roll

amplitude represented by the black curve gives therefore higher damping moments.

Another aspect can be seen from the graph: The distances between the single curves are

not equal. The damping moment increases not linear with the roll amplitude.

0

50

100

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0 1 2 3 4 5 6 7

Mo

men

t [N

m/r

ad]

Roll period [s]



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Ph

ase

of

mo

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eg]

Roll period [s]



6 / 9

The lower graph in figure 4 shows the phase of the moment relative to the rig motion in

degrees over the roll periods. For the first period of four seconds and for the large

periods towards the end, there is nearly no phase difference because either the water is

too inert to follow the rig motions or it can easily follow the motions at large periods. The

phase of the moment can be read out of the recorded time series after the measured

forces are multiplied by the lever arm to the centre of gravity.

FILLING LEVELS

The main dimensions of a ART are defined in design phase. Therfore, change the filling

percentage becomes the only way to ajust the natural frequency of the tank. In this part,

we will discuss three different filling levels: 5cm, 7.5cm and 10cm, corresponding to

33%, 50% and 66% filling percentage.

Figure 5. Moment comparison of different filling levels (Model scale)

Figure 6. Phase comparison of different filling levels (Model scale)

Normally, higher filling level means more liquid inside the tank which will provide more

static forece thus create more damping moment. This phenomenon can be found from 1s

to 2.5s in Figure 5.

0

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200

300

400

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600

0 1 2 3 4 5 6 7

Mo

men

t [N

m/r

ad]

Roll period [s]



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ase

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Roll period [s]



7 / 9

The natural period of the tank reduces as the filling level increases. It can be found that

higher filling level create higher peak moment.

When the sytem enters in inertial dominated region, the tank with more liquid generates

less moment. More liquid inside the tank means more mass, which creates more inertial

impact. Therefore, it is more difficult to excite the liquid motion. Besides, the system with

more mass enters in inertial dominated region much more early than those with less

mass. This can also be found in the phase curve.

When the phase close to 0 (after 3.0s), the system enters in interial dominated region.

Besides, it indicates the resonance occurs when the phase angle close to -90deg.

Then we transfer the model scale result to full scale result as shown in Figure 6.

Considering the resonance period of the vessel is around 13s (Figure 37), we can found

the tank performance at 50% filling is much better that it with 33% filling. The optimal

filling level for this particular tank is between 50% and 66%.

Figure 7. Comparison of different filling levels (Full scale) Figure 8. Roll RAO of the OSV

EXPERIMENTAL VALIDATION

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Mo

men

t [M

Nm

/rad

]

Roll period [s]


5cm filling height 7.5cm filling height 10cm filling height0

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4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

RA

O

RO

LL

η(4

)/A

[deg/m

]

WAVE PERIOD [sec]

DISPLACEMENTS

Project: Base Case - Operability

Base case ; 0.00kn 90.0°

8 / 9

Figure 9. Comparison between model test and simulation at period 6s

It can be found that the simulation has shown extremly good consistancy with the model

test at excitation period of 6s. Figure 51 and 52 illustrate the moment and phase

comparisons.

Figure 10. Moment and phase comparison between model test and simulation

0

5

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15

20

25

30

35

0 1 2 3 4 5 6

Mo

men

t [N

m]

Roll period [s]

Moment comparion [Nm]

Experiment CFD

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[deg

]

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Phase comparion [deg]

Experiment CFD

9 / 9

The simulations remain stable in stiffness dominated (1s-1.5s) and inertial dominated

(3s-6s) regions. Both moment and phase have shown high consistency in these regions.

The differences of the moment is less than 10% and the phase has a difference less than

6%.

However, in the damping dominated region (1.5s-3s), the difference between model test

and simulations should not be overlooked especially when the excitation period close to

the natural period of the tank where resonance occurs.

One of the reasons is the high courant number in the simulation. When the excitation

amplitude is 6deg, the motion and velocity is twice as much as that at 3deg amplitude.

Therefore, smaller time step is required to keep the courant numer at an acceptable

level. Therefore, the results will be more realistic and reliable. Another possible solution

to imporve the simulation result is to use 2nd-order temporal discretization scheme

instead of 1st-order scheme. Then we can keep the numerical diffusion at a lower level.

CONCLUSIONS

A numerical model is introduced and discussed. And it has been proved that this

particular model is suitable for a general thermostatic incompressible sloshing problem.

The main part of this project is studying the anti-roll tank performance. Different

excitation amplitudes have been investigated. It has been found that roll damping

increases nonlinear with increasing roll amplitude at resonance frequency region. Smaller

excitation amplitude will result in higher amping moment. However, the damping

moment has liner increase when the excitation period away from the damping dominated

region.

Besides, different filling levels are studied and discussed. It found that the roll damping

increases with increasing liquid level and the damping dominated region turned to be

more narrow. If known the natural period of vessel roll motion, the optimal filling level

can therefore be found.

Furthermore, it compares the results from simulations with the exsiting experimental

database. The comparions have shown good consistency. The model tests, which

designed and performed by the author, have given a strong validation for the numerical


Documents

ASTER THESIS - Bibsys Yue... · 2016-12-16 · In this thesis project, a numerical model based on Volume of Fluid ... Volume of Fluid (VOF), free surface tank (FST), model tests,